In Situ Characterization of Mixtures of Linear and Branched Hydrocarbons Confined within Porous Media Using 2D DQF-COSY NMR Spectroscopy

The analysis of 1D anti-diagonal spectra from the projections of 2D double-quantum filtered correlation spectroscopy NMR spectra is presented for the determination of the compositions of liquid mixtures of linear and branched alkanes confined within porous media. These projected spectra do not include the effects of line broadening and therefore retain high-resolution information even in the presence of inhomogeneous magnetic fields as are commonly found in porous media. A partial least-square regression analysis is used to characterize the mixture compositions. Two case studies are considered. First, mixtures of 2-methyl alkanes and n-alkanes are investigated. It is shown that estimation of the mol % of branched species present was achieved with a root-mean-square error of prediction (RMSEP) of 1.4 mol %. Second, the quantification of multicomponent mixtures consisting of linear alkanes and 2-, 3-, and 4-monomethyl alkanes was considered. Discrimination of 2-methyl and linear alkanes from other branched isomers in the mixture was achieved, although discrimination between 3- and 4- monomethyl alkanes was not possible. Compositions of the linear alkane, 2-methyl alkane, and the total composition of 3- and 4-methyl alkanes were estimated with a RMSEP <3 mol %. The approach was then used to estimate the composition of the mixtures in terms of submolecular groups of CH3CH2, (CH3)2CH, and CH2CH(CH3)CH2 present in the mixtures; a RMSEP <1 mol % was achieved for all groups. The ability to characterize the mixture compositions in terms of molecular subgroups allows the application of the method to characterize mixtures containing multimethyl alkanes. The motivation for this work is to develop a method for determining the mixture composition inside the catalyst pores during Fischer–Tropsch synthesis. However, the method reported is generic and can be applied to any system in which there is a need to characterize mixture compositions of linear and branched alkanes.


Table of contents
Section S1. Simulation of 1D anti-diagonal spectra of calibration mixtures ..

S1. Simulation of 1D anti-diagonal spectra of calibration mixtures
The calibration of PLSR models requires the spectral data X associated with mixtures of known compositions Y. In this work, the 1D anti-diagonal spectra of calibration mixtures were simulated by linear combination of the spectra of the individual species contained in the mixtures, according to the mixture composition. The simulation of mixture spectra is described as follows.
Given that the NMR signal of a species is proportional to the number of 1 H nuclear spins associated with that chemical species present in the mixture, the NMR data in the time domain, namely the free induction decay (FID), of a mixture is calculated as: where FIDmix is the simulated FID of the mixture of a reference number of moles, xi is the mole fraction of species i and FIDi is the FID of single-component species i of the reference number of moles. The experimentally measured FIDs of the single-component species will have different numbers of moles in the samples due to differing molar densities and sample volumes of the single-component samples. Therefore the experimentally measured FIDs were scaled to the reference number of moles using the known molar densities of the single components and the volume of the samples, where the volume was calculated from a 1D MRI profile of the sample. After obtaining FIDmix, the time domain data were processed to yield the 1D antidiagonal spectra as described in the main text.
The 1D anti-diagonal spectra of bulk liquid samples TM10-TM15 (Table S1) were simulated based on the mixture compositions and are compared with the spectra experimentally measured from these samples in Fig. S1. It is observed that in general the simulated spectra are in good agreement with the measured spectra with the largest error observed at Δδ = 0 ppm corresponding to the peak from the main diagonal in the 2D spectra. The relatively large error for this peak could be due to the fact that the suppression of signal for peaks on the main diagonal in the DQF-COSY experiment is sensitive to local magnetic field inhomogeneity which is slightly different for different samples. For cross peaks located at Δδ ≠ 0 ppm, the relative errors between the cross peak intensities of the simulated and measured spectra were calculated for the samples in Fig. S1 and an average value of these errors was 6%.

S2. Calibration of PLSR models
The PLSR method is introduced briefly as follows. In PLSR, the regression relationship between spectral matrix X (n × m, where n is the number of calibration samples and m is the number of data points in the anti-diagonal spectra) and composition matrix Y (n × p, where p is the number of mixture components or sub-molecular groups) is identified by finding the principal components (PCs) of X and Y with the covariance between these PCs maximised 1,2 .
The model structure of PLSR can be written as: where T (n × b, b is the number of PCs for X) is the score matrix of X, P (b × m) and Q (b × p) are the loading matrices of X and Y respectively, and E and F are the residual matrices. The score and loading matrices are obtained in model calibration. Estimation of the composition ̂ with the spectra Xtest of test samples is achieved as follows: where ̂ is the matrix of regression parameters, obtained using the score and loading matrices.
The PLSR models were calibrated using the calibration spectral data generated from the data of single-component bulk liquid of n-C12, 2-C7, 3-C7 and 4-C9 based on known mixture compositions following the method described in section S1. The spectral data and compositions of calibration mixtures constitute the matrices X and Y, and the creation of these two matrices are now described.
The composition matrix Y for a binary system was created with the composition of one The simulated spectra X and the corresponding compositions Y were then used to calibrate PLSR models. The calibration was implemented using the NIPALS algorithm 3 . The optimal numbers of principal components of the PLSR models were determined using the method of To calibrate PLSR models for estimating the compositions of groups CH3CH2, (CH3)2CH and CH2CH(CH3)CH2, the same spectral matrices X described earlier were used along with a group composition matrix Yg obtained from Y2 or Y4 as follows. The calculation of group compositions is based on molecular structures such that a n-C12 molecule contributes two CH3CH2 groups, a 2-C7 molecule contributes one (CH3)2CH group and one CH3CH2 group, S5 and a 4-C9 molecule contributes one CH2CH(CH3)CH2 group and two CH3CH2 groups. A 3-C7 molecule contributes one CH2CH(CH3)CH2 group and one CH3CH2 group; this is because the second CH3CH2 with carbon indices [1, 2'] does not contribute to cross peak intensity at  =  0.40 ppm (Table S2). In this work, only the CH3CH2 group associated with cross peaks at  =  0.40 ppm is considered. Denoting the composition vectors of n-C12, 2-C7, 3-C7 and 4-C9 as −C 12 , 2−C 7 , 3−C 7 and 4−C 9 respectively, the column of Yg corresponding to the CH3CH2 group that leads to the cross peaks at  =  0.40 ppm was calculated as where the multiplication constants of composition vectors indicate the number of CH3CH2 groups in a given molecule. The columns of Yg corresponding to (CH3)2CH and CH2CH(CH3)CH2 groups were obtained as 2−C 7 and 3−C 7 + 4−C 9 , respectively.

S3. Error analysis
The error of the PLSR estimation was evaluated using RMSE defined by: where N is the number of samples, ̂ and are the estimated and actual mole fractions respectively for sample i. When calculated for one sample N = 1, RMSE becomes the absolute error = |̂− |.
The standard error of the PLSR estimated compositions for each sample reported in Table 1 was calculated from 2-3 measurements of the same sample. Figure S1. Comparison between the simulated and measured 1D anti-diagonal spectra. The results for samples TM10-TM15 in Table S1 are presented in (a)-(f), respectively. The measured and simulated spectra are shown as black and blue, respectively. The difference between the simulated and measured spectra is shown in red.